A coin with heads facing up is shown as and a coin with tails facing up is…
2026
A coin with heads facing up is shown as
and a coin with tails facing up is shown as
.
Six coins are placed in the Starting Arrangement, as shown in the figure below. A “step” is defined as interchanging a pair of adjacent coins without flipping them. The minimum number of steps needed to go from the Starting Arrangement to the Final Arrangement, as shown in the figure, is ________.
Answer: C. 9 — Label the positions 1 to 6. The starting arrangement is H H H T T T and the target is T T T H H H. A single step swaps two adjacent coins, so it can only…
- A.
3
- B.
6
- C.
9
- D.
12
Attempted by 10 students.
Show answer & explanation
Correct answer: C
Label the positions 1 to 6. The starting arrangement is H H H T T T and the target is T T T H H H. A single step swaps two adjacent coins, so it can only exchange one H with one T that are next to each other. In the start, every H lies to the left of every T, giving 3 × 3 = 9 H‑before‑T pairs (inversions). In the target, no H is before a T, so the inversion count is 0. Each adjacent H‑T swap changes this count by exactly 1, and no swap of two equal coins helps, so at least 9 swaps are required. A valid 9‑step sequence is: H H H T T T → H H T H T T → H T H H T T → T H H H T T → T H H T H T → T H T H H T → T T H H H T → T T H H T H → T T H T H H → T T T H H H. Hence the minimum number of steps is 9.